Multi-peaked localized states of DNLS in one and two dimensions

Multi-peaked localized stationary solutions of the discrete nonlinear Schrodinger (DNLS) equation are presented in one (1D) and two (2D) dimensions. These are excited states of the discrete spectrum and correspond to multi-breather solutions. A simple, very fast, and efficient numerical method, suggested by Aubry, has been used for their calculation. The method involves no diagonalization, but just iterations of a map, starting from trivial solutions of the anti-continuous limit. Approximate analytical expressions are presented and compared with the numerical results. The linear stability of the calculated stationary states is discussed and the structure of the linear stability spectrum is analytically obtained for relatively large values of nonlinearity.Comment: 34 pages, 12 figure

Mechanical analogies for nonlinear light beams in nonlocal nematic liquid crystals

The equations governing nonlinear light beam propagation in nematic liquid crystals form a (2+1)-dimensional system consisting of a nonlinear Schrödinger-type equation for the electric field of the wavepacket and an elliptic equation for the reorientational response of the medium. The latter is "nonlocal" in the sense that it is much wider than the size of the beam. Due to these nonlocal, nonlinear features, there are no known general solutions of the nematic equations; hence, approximate methods have been found convenient to analyze nonlinear beam propagation in such media, particularly the approximation of solitary waves as mechanical particles moving in a potential. We review the use of dynamical equations to analyze solitary wave propagation in nematic liquid crystals through a number of examples involving their trajectory control, including comparisons with experimental results from the literature. Finally, we make a few general remarks on the existence and stability of optically self-localized solutions of the nematic equations

On non-local variational problems with lack of compactness related to non-linear optics

We give a simple proof of existence of solutions of the dispersion manage- ment and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local vari- ational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions' concentration compactness argument or Ekeland's variational principle.Comment: 30 page

Self-confined light waves in nematic liquid crystals

The study of light beams propagating in the nonlinear, dispersive, birefringent and nonlocal medium of nematic liquid crystals has attracted widespread interest in the last twenty years or so. We review hereby the underlying physics, theoretical modelling and numerical approximations for nonlinear beam propagation in planar cells filled with nematic liquid crystals, including bright and dark solitary waves, as well as optical vortices. The pertinent governing equations consist of a nonlinear Schrödinger-type equation for the light beam and an elliptic equation for the medium response. Since the nonlinear and coupled nature of this system presents difficulties in terms of finding exact solutions, we outline the various approaches used to resolve them, pinpointing the good agreement obtained with numerical solutions and experimental results. Measurement and material details complement the theoretical narration to underline the power of the modelling

Solitary waves in nematic liquid crystals

We study soliton solutions of a two-dimensional nonlocal NLS equation of Hartree-type with a Bessel potential kernel. The equation models laser propagation in nematic liquid crystals. Motivated by the experimental observation of spatially localized beams, see Conti et al. (2003), we show existence, stability, regularity, and radial symmetry of energy minimizing soliton solutions in R2. We also give theoretical lower bounds for the L2-norm (power) of these solitons, and show that small L2 -norm initial conditions lead to decaying solutions. We also present numerical computations of radial soliton solutions. These solutions exhibit the properties expected by the infinite plane theory, although we also see some finite (computational) domain effects, especially solutions with arbitrarily small power